Analog and Digital Signals and Systems (eBook)

R. K. Rao Yarlagadda (Autor)

eBook Download: PDF

2010 | 1. Auflage
555 Seiten
Springer US (Verlag)
978-1-4419-0034-0 (ISBN)

Infos

This book presents a systematic, comprehensive treatment of analog, discrete signal analysis and synthesis and an introduction to analog communication theory. The material is divided into five parts. The first part (Chapters 1-3) is mathematically oriented and deals with continuous-time (analog) signals. The second part (Chapters 4-5) is again mathematically based and deals with Fourier transforms, Hartley transforms, Laplace and Hilbert transforms. The third part (Chapters 6-7) presents basic system analysis, approximations, and analog filter circuits using op-amps. The fourth part (Chapters 8-9) deals with discrete signals, fast algorithms and digital filters. The fifth part deals with an introduction to analog communication systems.

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This book presents a systematic, comprehensive treatment of analog and discrete signal analysis and synthesis and an introduction to analog communication theory. This evolved from my 40 years of teaching at Oklahoma State University (OSU). It is based on three courses, Signal Analysis (a second semester junior level course), Active Filters (a first semester senior level course), and Digital signal processing (a second semester senior level course). I have taught these courses a number of times using this material along with existing texts. The references for the books and journals (over 160 references) are listed in the bibliography section. At the undergraduate level, most signal analysis courses do not require probability theory. Only, a very small portion of this topic is included here. I emphasized the basics in the book with simple mathematics and the soph- tication is minimal. Theorem-proof type of material is not emphasized. The book uses the following model: 1. Learn basics 2.Check the work using bench marks 3. Use software to see if the results are accurate The book provides detailed examples (over 400) with applications. A thr- number system is used consisting of chapter number - section number - example or problem number, thus allowing the student to quickly identify the related material in the appropriate section of the book. The book includes well over 400 homework problems. Problem numbers are identified using the above three-number system.

Inhalt

Analog and Digital Signals and Systems 1
Note to Instructors 5
Preface 6
Summary of the Chapters 6
Suggested Course Content 7
Acknowledgements 9
Contents 10
List of Tables 22
Basic Concepts in Signals 24
1.1 Introduction to the Book and Signals 24
1.1.1 Different Ways of Looking at a Signal 24
1.1.2 Continuous-Time and Discrete-Time Signals 26
1.1.3 Analog Versus Digital Signal Processing 28
1.1.4 Examples of Simple Functions 29
1.2 Useful Signal Operations 31
1.2.1 Time Shifting 31
1.2.2 Time Scaling 31
1.2.3 Time Reversal 31
1.2.4 Amplitude Shift 31
1.2.5 Simple Symmetries: Even and Odd Functions 32
1.2.6 Products of Even and Odd Functions 32
1.2.7 Signum (or sgn) Function 33
1.2.8 Sinc and Sinc2 Functions 33
1.2.9 Sine Integral Function 33
1.3 Derivatives and Integrals of Functions 34
1.3.1 Integrals of Functions with Symmetries 35
1.3.2 Useful Functions from Unit Step Function 35
1.3.3 Leibniz’s Rule 36
1.3.4 Interchange of a Derivative and an Integral 36
1.3.5 Interchange of Integrals 36
1.4 Singularity Functions 37
1.4.1 Unit Impulse as the Limit of a Sequence 38
1.4.2 Step Function and the Impulse Function 39
1.4.3 Functions of Generalized Functions 40
1.4.4 Functions of Impulse Functions 41
1.4.5 Functions of Step Functions 42
1.5 Signal Classification Based on Integrals 42
1.5.1 Effects of Operations on Signals 44
1.5.2 Periodic Functions 44
1.5.3 Sum of Two Periodic Functions 46
1.6 Complex Numbers, Periodic, and Symmetric Periodic Functions 47
1.6.1 Complex Numbers 48
1.6.2 Complex Periodic Functions 50
1.6.3 Functions of Periodic Functions 50
1.6.4 Periodic Functions with Additional Symmetries 51
1.7 Examples of Probability Density Functions and their Moments 52
1.8 Generation of Periodic Functions from Aperiodic Functions 54
1.9 Decibel 55
1.10 Summary 57
Problems 58
Convolution and Correlation 61
2.1 Introduction 61
2.1.1 Scalar Product and Norm 62
2.2 Convolution 63
2.2.1 Properties of the Convolution Integral 63
2.2.2 Existence of the Convolution Integral 66
2.3 Interesting Examples 66
2.4 Convolution and Moments 72
2.4.1 Repeated Convolution and the Central Limit Theorem 74
2.4.2 Deconvolution 75
2.5 Convolution Involving Periodic and Aperiodic Functions 76
2.5.1 Convolution of a Periodic Function with an Aperiodic Function 76
2.5.2 Convolution of Two Periodic Functions 77
2.6 Correlation 78
2.6.1 Basic Properties of Cross-Correlation Functions 79
2.6.2 Cross-Correlation and Convolution 79
2.6.3 Bounds on the Cross-Correlation Functions 80
2.6.4 Quantitative Measures of Cross-Correlation 81
2.7 Autocorrelation Functions of Energy Signals 85
2.8 Cross- and Autocorrelation of Periodic Functions 87
2.9 Summary 90
Problems 90
Fourier Series 92
3.1 Introduction 92
3.2 Orthogonal Basis Functions 93
3.2.1 Gram-Schmidt Orthogonalization 95
3.3 Approximation Measures 96
3.3.1 Computation of c[k] Based on Partials 98
3.3.2 Computation of c[k] Using the Method of Perfect Squares 98
3.3.3 Parseval’s Theorem 99
3.4 Fourier Series 101
3.4.1 Complex Fourier Series 101
3.4.2 Trigonometric Fourier Series 104
3.4.3 Complex F-series and the Trigonometric F-series Coefficients-Relations 104
3.4.4 Harmonic Form of Trigonometric Fourier Series 104
3.4.5 Parseval’s Theorem Revisited 105
3.4.6 Advantages and Disadvantages of the Three Forms of Fourier Series 106
3.5 Fourier Series of Functions with Simple Symmetries 106
3.5.1 Simplification of the Fourier Series Coefficient Integral 107
3.6 Operational Properties of Fourier Series 108
3.6.1 Principle of Superposition 108
3.6.2 Time Shift 108
3.6.3 Time and Frequency Scaling 109
3.6.4 Fourier Series Using Derivatives 110
3.6.5 Bounds and Rates of Fourier Series Convergence by the Derivative Method 112
3.6.6 Integral of a Function and Its Fourier Series 114
3.6.7 Modulation in Time 114
3.6.8 Multiplication in Time 115
3.6.9 Frequency Modulation 116
3.6.10 Central Ordinate Theorems 116
3.6.11 Plancherel’s Relation (or Theorem) 116
3.6.12 Power Spectral Analysis 116
3.7 Convergence of the Fourier Series and the Gibbs Phenomenon 117
3.7.1 Fourier’s Theorem 117
3.7.2 Gibbs Phenomenon 118
3.7.3 Spectral Window Smoothing 120
3.8 Fourier Series Expansion of Periodic Functions with Special Symmetries 121
3.8.1 Half-Wave Symmetry 121
3.8.2 Quarter-Wave Symmetry 123
3.8.3 Even Quarter-Wave Symmetry 123
3.8.4 Odd Quarter-Wave Symmetry 123
3.8.5 Hidden Symmetry 124
3.9 Half-Range Series Expansions 124
3.10 Fourier Series Tables 125
3.11 Summary 125
Problems 127
Fourier Transform Analysis 130
4.1 Introduction 130
4.2 Fourier Series to Fourier Integral 130
4.2.1 Amplitude and Phase Spectra 133
4.2.2 Bandwidth-Simplistic Ideas 135
4.3 Fourier Transform Theorems, Part 1 135
4.3.1 Rayleigh’s Energy Theorem 135
4.3.2 Superposition Theorem 136
4.3.3 Time Delay Theorem 137
4.3.4 Scale Change Theorem 137
4.3.5 Symmetry or Duality Theorem 139
4.3.6 Fourier Central Ordinate Theorems 140
4.4 Fourier Transform Theorems, Part 2 140
4.4.1 Frequency Translation Theorem 141
4.4.2 Modulation Theorem 141
4.4.3 Fourier Transforms of Periodic and Some Special Functions 142
4.4.4 Time Differentiation Theorem 145
4.4.5 Times-t Property: Frequency Differentiation Theorem 147
4.4.6 Initial Value Theorem 149
4.4.7 Integration Theorem 149
4.5 Convolution and Correlation 150
4.5.1 Convolution in Time 150
4.5.2 Proof of the Integration Theorem 153
4.5.3 Multiplication Theorem (Convolution in Frequency) 154
4.5.4 Energy Spectral Density 156
4.6 Autocorrelation and Cross-Correlation 157
4.6.1 Power Spectral Density 159
4.7 Bandwidth of a Signal 160
4.7.1 Measures Based on Areas of the Time and Frequency Functions 160
4.7.2 Measures Based on Moments 161
4.7.3 Uncertainty Principle in Fourier Analysis 162
4.8 Moments and the Fourier Transform 164
4.9 Bounds on the Fourier Transform 165
4.10 Poisson’s Summation Formula 166
4.11 Interesting Examples and a Short Fourier Transform Table 166
4.11.1 Raised-Cosine Pulse Function 167
4.12 Tables of Fourier Transforms Properties and Pairs 168
4.13 Summary 168
Problems 168
Relatives of Fourier Transforms 175
5.1 Introduction 175
5.2 Fourier Cosine and Sine Transforms 176
5.3 Hartley Transform 179
5.4 Laplace Transforms 181
5.4.1 Region of Convergence (ROC) 183
5.4.2 Inverse Transform of Two-Sided Laplace Transform 184
5.4.3 Region of Convergence (ROC) of Rational Functions - Properties 185
5.5 Basic Two-Sided Laplace Transform Theorems 185
5.5.1 Linearity 185
5.5.2 Time Shift 185
5.5.3 Shift in s 185
5.5.4 Time Scaling 185
5.5.5 Time Reversal 186
5.5.6 Differentiation in Time 186
5.5.7 Integration 186
5.5.8 Convolution 186
5.6 One-Sided Laplace Transform 186
5.6.1 Properties of the One-Sided Laplace Transform 187
5.6.2 Comments on the Properties (or Theorems) of Laplace Transforms 187
5.7 Rational Transform Functions and Inverse Laplace Transforms 194
5.7.1 Rational Functions, Poles, and Zeros 195
5.7.2 Return to the Initial and Final Value Theorems and Their Use 196
5.8 Solutions of Constant Coefficient Differential Equations Using Laplace Transforms 198
5.8.1 Inverse Laplace Transforms 199
5.8.2 Partial Fraction Expansions 199
5.9 Relationship Between Laplace Transforms and Other Transforms 203
5.9.1 Laplace Transforms and Fourier Transforms 204
5.9.2 Hartley Transforms and Laplace Transforms 205
5.10 Hilbert Transform 206
5.10.1 Basic Definitions 206
5.10.2 Hilbert Transform of Signals with Non-overlapping Spectra 208
5.10.3 Analytic Signals 209
5.11 Summary 210
Problems 210
Systems and Circuits 213
6.1 Introduction 213
6.2 Linear Systems, an Introduction 213
6.3 Ideal Two-Terminal Circuit Components and Kirchhoff ’s Laws 214
6.3.1 Two-Terminal Component Equations 215
6.3.2 Kirchhoff’s Laws 217
6.4 Time-Invariant and Time-Varying Systems 218
6.5 Impulse Response 219
6.5.1 Eigenfunctions 222
6.5.2 Bounded-Input/Bounded-Output (BIBO) Stability 222
6.5.3 Routh-Hurwitz Criterion (R-H criterion) 223
6.5.4 Eigenfunctions in the Fourier Domain 226
6.6 Step Response 228
6.7 Distortionless Transmission 233
6.7.1 Group Delay and Phase Delay 233
6.8 System Bandwidth Measures 236
6.8.1 Bandwidth Measures Using the Impulse Response $ /curr h({/rm t})$ and Its Transform $/curr{ H({/rm j}/omega )}$ 236
6.8.2 Half-Power or 3 dB Bandwidth 237
6.8.3 Equivalent Bandwidth or Noise Bandwidth 237
6.8.4 Root Mean-Squared (RMS) Bandwidth 238
6.9 Nonlinear Systems 239
6.9.1 Distortion Measures 240
6.9.2 Output Fourier Transform of a Nonlinear System 240
6.9.3 Linearization of Nonlinear System Functions 241
6.10 Ideal Filters 241
6.10.1 Low-Pass, High-Pass, Band-Pass, and Band-Elimination Filters 242
6.11 Real and Imaginary Parts of the Fourier Transform of a Causal Function 247
6.11.1 Relationship Between Real and Imaginary Parts of the Fourier Transform of a Causal Function Using Hilbert Transform 248
6.11.2 Amplitude Spectrum |H(jw)| to a Minimum Phase Function H(s) 249
6.12 More on Filters: Source and Load Impedances 249
6.12.1 Simple Low-Pass Filters 251
6.12.2 Simple High-Pass Filters 251
6.12.3 Simple Band-Pass Filters 253
6.12.4 Simple Band-Elimination or Band-Reject or Notch Filters 255
6.12.5 Maximum Power Transfer 258
6.12.6 A Simple Delay Line Circuit 259
6.13 Summary 259
Problems 260
Approximations and Filter Circuits 263
7.1 Introduction 263
7.2 Bode Plots 266
7.2.1 Gain and Phase Margins 272
7.3 Classical Analog Filter Functions 274
7.3.1 Amplitude-Based Design 274
7.3.2 Butterworth Approximations 275
7.3.3 Chebyshev (Tschebyscheff) Approximations 277
7.4 Phase-Based Design 282
7.4.1 Maximally Flat Delay Approximation 283
7.4.2 Group Delay of Bessel Functions 284
7.5 Frequency Transformations 286
7.5.1 Normalized Low-Pass to High-Pass Transformation 286
7.5.2 Normalized Low-Pass to Band-Pass Transformation 288
7.5.3 Normalized Low-Pass to Band-Elimination Transformation 288
7.5.4 Conversions of Specifications from Low-Pass, High-Pass, Band-Pass, and Band Elimination Filters to Normalized Low-Pass Filters 290
7.6 Multi-terminal Components 293
7.6.1 Two-Port Parameters 293
7.6.2 Circuit Analysis Involving Multi-terminal Components and Networks 297
7.6.3 Controlled Sources 298
7.7 Active Filter Circuits 299
7.7.1 Operational Amplifiers, an Introduction 299
7.7.2 Inverting Operational Amplifier Circuits 300
7.7.3 Non-inverting Operational Amplifier Circuits 302
7.7.4 Simple Second-Order Low-Pass and All-Pass Circuits 304
7.8 Gain Constant Adjustment 305
7.9 Scaling 307
7.9.1 Amplitude (or Magnitude) Scaling, RLC Circuits 307
7.9.2 Frequency Scaling, RLC Circuits 308
7.9.3 Amplitude and Frequency Scaling in Active Filters 308
7.9.4 Delay Scaling 310
7.10 RC-CR Transformations: Low-Pass to High-Pass Circuits 312
7.11 Band-Pass, Band-Elimination and Biquad Filters 314
7.12 Sensitivities 318
7.13 Summary 321
Problems 321
Discrete-Time Signals and Their Fourier Transforms 330
8.1 Introduction 330
8.2 Sampling of a Signal 331
8.2.1 Ideal Sampling 331
8.2.2 Uniform Low-Pass Sampling or the Nyquist Low-Pass Sampling Theorem 333
8.2.3 Interpolation Formula and the Generalized Fourier Series 336
8.2.4 Problems Associated with Sampling Below the Nyquist Rate 338
8.2.5 Flat Top Sampling 341
8.2.6 Uniform Band-Pass Sampling Theorem 343
8.2.7 Equivalent continuous-time and discrete-time systems 344
8.3 Basic Discrete-Time (DT) Signals 344
8.3.1 Operations on a Discrete Signal 346
8.3.2 Discrete-Time Convolution and Correlation 348
8.3.3 Finite duration, right-sided, left-sided, two-sided, and causal sequences 349
8.3.4 Discrete-Time Energy and Power Signals 349
8.4 Discrete-Time Fourier Series 351
8.4.1 Periodic Convolution of Two Sequences with the Same Period 353
8.4.2 Parseval’s Identity 353
8.5 Discrete-Time Fourier Transforms 354
8.5.1 Discrete-Time Fourier Transforms (DTFTs) 354
8.5.2 Discrete-Time Fourier Transforms of Real Signals with Symmetries 355
8.6 Properties of the Discrete-Time Fourier Transforms 358
8.6.1 Periodic Nature of the Discrete-Time Fourier Transform 358
8.6.2 Superposition or Linearity 359
8.6.3 Time Shift or Delay 360
8.6.4 Modulation or Frequency Shifting 360
8.6.5 Time Scaling 360
8.6.6 Differentiation in Frequency 361
8.6.7 Differencing 361
8.6.8 Summation or Accumulation 363
8.6.9 Convolution 363
8.6.10 Multiplication in Time 364
8.6.11 Parseval’s Identities 365
8.6.12 Central Ordinate Theorems 365
8.6.13 Simple Digital Encryption 365
8.7 Tables of Discrete-Time Fourier Transform (DTFT) Properties and Pairs 366
8.8 Discrete-Time Fourier-transforms from Samples of the Continuous-Time Fourier-Transforms 367
8.9 Discrete Fourier Transforms (DFTs) 369
8.9.1 Matrix Representations of the DFT and the IDFT 371
8.9.2 Requirements for Direct Computation of the DFT 372
8.10 Discrete Fourier Transform Properties 373
8.10.1 DFTs and IDFTs of Real Sequences 373
8.10.2 Linearity 373
8.10.3 Duality 374
8.10.4 Time Shift 374
8.10.5 Frequency Shift 375
8.10.6 Even Sequences 375
8.10.7 Odd Sequences 375
8.10.8 Discrete-Time Convolution Theorem 376
8.10.9 Discrete-Frequency Convolution Theorem 377
8.10.10 Discrete-Time Correlation Theorem 378
8.10.11 Parseval’s Identity or Theorem 378
8.10.12 Zero Padding 378
8.10.13 Signal Interpolation 379
8.10.14 Decimation 381
8.11 Summary 381
Problems 381
Discrete Data Systems 385
9.1 Introduction 385
9.2 Computation of Discrete Fourier Transforms (DFTs) 386
9.2.1 Symbolic Diagrams in Discrete-Time Representations 386
9.2.2 Fast Fourier Transforms (FFTs) 387
9.3 DFT (FFT) Applications 390
9.3.1 Hidden Periodicity in a Signal 390
9.3.2 Convolution of Time-Limited Sequences 392
9.3.3 Correlation of Discrete Signals 395
9.3.4 Discrete Deconvolution 396
9.4 z-Transforms 398
9.4.1 Region of Convergence (ROC) 399
9.4.2 z-Transform and the Discrete-Time Fourier Transform (DTFT) 402
9.5 Properties of the z-Transform 402
9.5.1 Linearity 402
9.5.2 Time-Shifted Sequences 403
9.5.3 Time Reversal 403
9.5.4 Multiplication by an Exponential 403
9.5.5 Multiplication by n 404
9.5.6 Difference and Accumulation 404
9.5.7 Convolution Theorem and the z-Transform 404
9.5.8 Correlation Theorem and the z-Transform 405
9.5.9 Initial Value Theorem in the Discrete Domain 406
9.5.10 Final Value Theorem in the Discrete Domain 406
9.6 Tables of z-Transform Properties and Pairs 407
9.7 Inverse z-Transforms 408
9.7.1 Inversion Formula 408
9.7.2 Use of Transform Tables (Partial Fraction Expansion Method) 409
9.7.3 Inverse z-Transforms by Power Series Expansion 412
9.8 The Unilateral or the One-Sided z-Transform 413
9.8.1 Time-Shifting Property 413
9.9 Discrete-Data Systems 415
9.9.1 Discrete-Time Transfer Functions 418
9.9.2 Schur-Cohn Stability Test 419
9.9.3 Bilinear Transformations 419
9.10 Designs by the Time and Frequency Domain Criteria 421
9.10.1 Impulse Invariance Method by Using the Time Domain Criterion 423
9.10.2 Bilinear Transformation Method by Using the Frequency Domain Criterion 425
9.11 Finite Impulse Response (FIR) Filter Design 428
9.11.1 Low-Pass FIR Filter Design 429
9.11.2 High-Pass, Band-Pass, and Band-Elimination FIR Filter Designs 431
9.11.3 Windows in Fourier Design 434
9.12 Digital Filter Realizations 437
9.12.1 Cascade Form of Realization 440
9.12.2 Parallel Form of Realization 440
9.12.3 All-Pass Filter Realization 441
9.12.4 Digital Filter Transposed Structures 441
9.12.5 FIR Filter Realizations 441
9.13 Summary 442
Problems 443
Analog Modulation 446
10.1 Introduction 446
10.2 Limiters and Mixers 448
10.2.1 Mixers 449
10.3 Linear Modulation 449
10.3.1 Double-Sideband (DSB) Modulation 449
10.3.2 Demodulation of DSB Signals 450
10.4 Frequency Multipliers and Dividers 452
10.5 Amplitude Modulation (AM) 454
10.5.1 Percentage Modulation 455
10.5.2 Bandwidth Requirements 455
10.5.3 Power and Efficiency of an Amplitude Modulated Signal 456
10.5.4 Average Power Contained in an AM Signal 457
10.6 Generation of AM Signals 458
10.6.1 Square-Law Modulators 458
10.6.2 Switching Modulators 458
10.6.3 Balanced Modulators 459
10.7 Demodulation of AM Signals 460
10.7.1 Rectifier Detector 460
10.7.2 Coherent or a Synchronous Detector 460
10.7.3 Square-Law Detector 461
10.7.4 Envelope Detector 461
10.8 Asymmetric Sideband Signals 463
10.8.1 Single-Sideband Signals 463
10.8.2 Vestigial Sideband Modulated Signals 464
10.8.3 Demodulation of SSB and VSB Signals 465
10.8.4 Non-coherent Demodulation of SSB 466
10.8.5 Phase-Shift Modulators and Demodulators 466
10.9 Frequency Translation and Mixing 467
10.10 Superheterodyne AM Receiver 470
10.11 Angle Modulation 472
Chap10Sec31 473
10.12.1 Narrowband (NB) Angle Modulation 475
10.12.2 Generation of Angle Modulated Signals 476
10.12 Spectrum of an Angle Modulated Signal 477
10.12.1 Properties of Bessel Functions 478
10.12.2 Power Content in an Angle Modulated Signal 480
10.13 Demodulation of Angle Modulated Signals 482
10.13.1 Frequency Discriminators 482
10.13.2 Delay Lines as Differentiators 484
10.14 FM Receivers 485
10.14.1 Distortions 485
10.14.2 Pre-emphasis and De-emphasis 486
10.14.3 Distortions Caused by Multipath Effect 487
10.15 Frequency-Division Multiplexing (FDM) 488
10.15.1 Quadrature Amplitude Modulation (QAM) or Quadrature Multiplexing (QM) 489
10.15.2 FM Stereo Multiplexing and the FM Radio 490
10.16 Pulse Modulations 491
10.16.1 Pulse Amplitude Modulation (PAM) 492
10.16.2 Problems with Pulse Modulations 492
10.16.3 Time-Division Multiplexing (TDM) 494
10.17 Pulse Code Modulation (PCM) 495
10.17.1 Quantization Process 495
10.17.2 More on Coding 497
10.17.3 Tradeoffs Between Channel Bandwidth and Signal-to-Quantization Noise Ratio 498
10.17.4 Digital Carrier Modulation 499
10.18 Summary 501
Problems 501
Appendix A: Matrix Algebra 505
A1 Matrix Notations 505
A.2 Elements of Matrix Algebra 506
A.2.1 Vector Norms 507
A.3 Solutions of Matrix Equations 508
A.3.1 Determinants 508
A.3.2 Cramer’s Rule 509
A.3.3 Rank of a Matrix 510
A.4 Inverses of Matrices and Their Use in Determining the Solutions of a Set of Equations 511
A.5 Eigenvalues and Eigenvectors 512
A.6 Singular Value Decomposition (SVD) 516
A.7 Generalized Inverses of Matrices 517
A.8 Over- and Underdetermined System of Equations 518
A. 8.1 Least-Squares Solutions of Overdetermined System of Equations (m> n)
A.8.2 Least-Squares Solution of Underdetermined System of Equations (< m<
A.9 Numerical-Based Interpolations: Polynomial and Lagrange Interpolations 521
A.9.1 Polynomial Approximations 521
A.9.2 Lagrange Interpolation Formula 522
Problems 522
Appendix B: MATLAB1 for Digital Signal Processing 525
B.1 Introduction 525
B.2 Signal Representation 525
B.3 Signal Integration 527
B.4 Fast Fourier Transforms (FFTs) 527
B.5 Convolution of Signals 529
B.6 Differentiation Using Numerical Methods 531
B.7 Fourier Series Computation 531
B.8 Roots of Polynomials, Partial Fraction Expansions, Pole-Zero Functions 533
B.8.1 Partial Fraction Expansions 534
B.9 Bode Plots, Impulse and Step Responses 534
B.9.1 Bode Plots 534
B.9.2 Impulse and Step Responses 534
B.10 Frequency Responses of Digital Filter Transfer Functions 536
B.11 Introduction to the Construction of Simple MATLAB Functions 536
B.12 Additional MATLAB Code 537
Appendix C: Mathematical Relations 539
C.1 Trigonometric Identities 539
C.2 Logarithms, Exponents and Complex Numbers 539
C.3 Derivatives 540
C.4 Indefinite Integrals 540
C.5 Definite Integrals and Useful Identities 541
C.6 Summation Formulae 541
C.7 Series Expansions 542
C.8 Special Constants and Factorials 542
Bibliography 543
Author Index 547
Subject Index 550

Produkt-Details

• Erscheint lt. Verlag: 5.8.2010
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• ISBN-10: 1-4419-0034-9 / 1441900349
• ISBN-13: 978-1-4419-0034-0 / 9781441900340